Armstrong's axioms are a set of inference rules used to derive additional functional dependencies (FDs) from a given set of FDs in Relational Database Management Systems (RDBMS). These axioms were proposed by William W. Armstrong and are a foundation for understanding and reasoning about FDs. There are three main axioms:

1. Reflexivity:

If Y is a subset of X, then X → Y holds. This axiom states that if a set of attributes X functionally determines another set of attributes Y, then X also functionally determines any subset of Y.

2. Augmentation:

If X → Y holds, then XZ → YZ holds for any set of attributes Z. This axiom states that if a set of attributes X functionally determines another set of attributes Y, then adding the same set of attributes Z to both X and Y preserves the functional dependency.

3. Transitivity:

If X → Y and Y → Z hold, then X → Z holds. This axiom states that if a set of attributes X functionally determines another set of attributes Y and Y functionally determines another set of attributes Z, then X functionally determines Z.

Using these axioms, additional FDs can be derived through a process called Armstrong's closure. The closure of a set of attributes F, denoted as F+, is the set of all FDs that can be logically inferred from the given set of FDs F using Armstrong's axioms.

Armstrong's axioms provide a theoretical foundation for understanding and manipulating FDs in RDBMS. They enable the identification and derivation of additional FDs, which is crucial for database design, normalization, and ensuring data integrity.

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